Archive for the ‘mathematics’ Category
[ by Charles Cameron — infernal and celestial geometries ]
It is well known that the Platonic “ideal” circle is not to be found in the “real” world of people and things, since it would be composed of an infinte number of non-dimensional points. Human and inhumane circles, however, are another matter.
The upper circle shows spectators who gathered around the mangled body of an alleged homosexual, thrown from the roof of a seven-storey building by members of the Islamic State, and stoned the still breathing victim to death.
The lower circle is intended as a counter-weight to the atrocity shown above it. It is the Zen master Hakuin’s enso or zen brush-and-ink circle, perfect in its imperfection, its human spontaneity, and certainly not in the Platonic “ideal” sense.
The Topological Musings blog quotes Plato from The Republic, deftly avoiding any mention of circles…
And do you not also know that they (mathematicians) further make use of the visible forms and talk about them, though they are not thinking of them but of those things of which they are a likeness, pursuing their inquiry for the sake of the square as such and the diagonal as such, and not for the sake of the image of it which they draw?… The very things which they mold and draw, … they treat in their turn as only images, but what they really seek is to get sight of those realities which can be seen only by the mind.
Three circles: the utterly inhumane, the perfectly imperfect, and the impossible.
For a “transgressive” study of the issue of homosexuality and the seventh circle of Dante’s Inferno, see John Boswell‘s Dante and the Sodomites, in Dante Studies, No. 112 (1994), pp. 63-76. What exactly “transgressive” means, I have yet to understand. I do however, personally, abhor people throwing other people off high buildings and / or stoning them to death.
[ by Charles Cameron — the geometry of two miracle stories from Abdullah Azzam ]
These two tales are taken from Abdullah Azzam, Signs of ar-Rahman in the Jihad of Afghanistan.
Binary oppositions seem to be basic to the human thought process, and this simple, elegant observation has in turn given rise to a number of interesting philosopphical explorations, some of which are expressed perhaps most powerully in diagrams. I am thinking here of the medieval square of opposition — as in this diagram taken from Georg Reisch, Margarita Phylosophica tractans de omni genere scibili, Basel 1517:
Algirdas Greimas developed his semiotic square from this medieval diagram —
— and defines his square as the “visual representation of the logical articulation of any category”. In his “Towards a Theory of Modalities”, Greimas writes:
the terms manifestation vs. immanence .. can be compared profitably with the categories surface vs. deep in linguistics, manifest vs. latent in psychoanalysis, phenomenal vs. noumenal in philosophy, etc.
Then there’s Levi-Strauss and his triangle, essentially a variant on the same idea, applied by LS in his magnificent 4-volume Mythologiques to a wide range of myths — here’s the basic triangle for the first volume, The Raw and the Cooked:
What Reisch, Greimas and Levi-Strauss are all doing lies in its own distinct area of “visual thinking” at the confluence of logic, algebra, geometry and conceptual graphs — the same area my own DoubleQuotes and the HipBone and Sembl games are found in.
When people think about narrative — and it is or should be as hot a topic in strategy and counterterrorism as it is in myth, story-telling, film and their various related forms of criticism — they tend to think linearly, from beginning to end, noting the emotional expansions and contractions, the narrative shifts, the crescendos before the climax and its resolution.
My own style of thinking leans more to the atemporal or synchronic, which in turn is closer to the logical-algebraic-geometric-graphical mode of visual expression. Thus, for me, the “myth of Narcissus” is not a story-line but a geometry, a narrative formulation of the concept of reflection, or “bouncing back”. To adapt the Levi-Strauss triangle to the Narcissus narrative, then, we have:
while the two Azzam miracle tales in my DoubleQuote at the top of this post give us:
This in turn can become a square if we allow the four coordinates to be wine (intoxicant, bad), water (sobriety, good), vinegar (sour, bad) and honey (sweet, good). We notice here that water (sobriety, good) is the fourth which hovers unmentioned over the twin tales, just as Jung argued the dogma of the Assumption of the Virgin into heaven was the “fourth” which “completed” — nb, this is from a psychological perspective — the celestial Trinity of Father, Son, and Holy Spirit.
It remains for Jalaluddin Rumi to transcend the duality of the halal (sobriety) and the haram (intoxication) in his praise of his master, Shams of Tabriz:
In Shams al-Din-i Tabrizi you will discover a heart which is at once intoxicated and very sober.
In what sense or senses are Azzam’s two tales two, and in what sense are they one and the same?
Sources & suggested further readings include:
The Raw and the Cooked: Mythologiques, Volume 1 Anthropology for Beginners Structure, Sign, and Play in the Discourse of the Human Sciences The Dual and the Real Semiotics for Beginners Semiotics and Language Visual Memory (handbags!) Punctualization: Law and Greimas Square of Opposition Visualizing knowledge Signs of Ar-Rahman Mystical Poems of Rumi
[ by Charles Cameron — who believes that detours are the spice of life ]
There’s a fasacinating article about Craig Kaplan and his work with tiling that I came across today, Crazy paving: the twisted world of parquet deformations — I highly recommend it to anyone interested in pattern — and I highly recommend anyone uninterested in pattern to get interested!
Kaplan himself is no stranger to Escher’s work, obviously enough — he’s even written a paper, Metamorphosis in Escher’s Art — the abstract reads:
M.C. Escher returned often to the themes of metamorphosis and deformation in his art, using a small set of pictorial devices to express this theme. I classify Escher’s various approaches to metamorphosis, and relate them to the works in which they appear. I also discuss the mathematical challenges that arise in attempting to formalize one of these devices so that it can be applied reliably.
I mean Kaplan no dishonor, then, when I say that his algorithmic tilings, as seen in the upper panel above, still necessarily lack something that his mentor’s images have, as seen in the lower panel — a quirky willingness to go beyond pattern into a deeper pattern, as when the turreted outcropping of a small Italian town on the Amalfi coast becomes a rook in the game of chess…
Comparing one with the other, I am reminded of the differences between quantitative and qualitative approaches to understanding, of SIGINT and HUMINT in terms of the types of intelligence collected — and at the philosophical limit, of the very notions of quantity and quality.
[ by Charles Cameron — two approaches to mathematics, two types of heroism, and their respective complementarities ]
I wish to propose a clear analogy between the mathematician Grothendieck‘s two styles of approach to a problem in mathematics, and the Navajo Twin Gods, Monster-Slayer and Child-Born-of-Water.
Steve Landsburg‘s post, The Generalist, compares two approaches to mathematics, as practiced by two eminent mathematicians:
If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”
In the paras leading up to this one, Landsburg gives us the insight that these two approaches can be generalized as “zooming in” and “zooming out”:
Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.
Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.
It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to forget all about the inner workings of clocks and ask “What else falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.
Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…
Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is all numbers, you’ve got a good chance of figuring out why they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.
Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.
I no longer have — alas — a copy of Where the Two Came to their Father, the first volume in the Bollingen Series, with its suite of 18 sand paintings beautifully rendered in silkscreen by Maud Oakes, but their respective black and blue colorations lead me to suppose that the illustration at the head of this post, taken rom that series, shows the twin heroes, Monster Slayer (black) and Child Born of Water (blue) whose journeys and initiation are the subject of the rituasl “sing” recorded in that book.
The theme of two male hero twins is central to the mythologies of the American continent, according to Jospeh Campbell, who contributed a commentary to Oakes’ recording of Jeff King‘s performance of this ceremony, and lacking both the King > Oakes > Campbell book and Gladys Reichard‘s two volumes on Navaho Religion, I must draw on brief quotes from miscellaneous web sources to dramatize the differences between the twins.
Monster Slayer is the doer of deeds, similar in nature to other masculine, not to say macho, heroes — while Child Born of Water is the contemplative of the pair:
The Sun [Jóhonaa’éí] gave them prayersticks and then told them that the younger of the two (Born for Water) would sit watching these prayersticks while the older (Monster Slayer) went out to kill the monsters. If these prayersticks began to burn, this would signal that his brother was in danger and that he should go to him to help.
Monster Slayer (na’ye’ ne’zyani) (I) represents impulsive aggression, whereas Child-of-the-water represents reserve, caution, and thoughtful preparation.
A measure of their respective strategies, and of the ways in which the insights of Child Born of Water can succeed where the brute force tactics of Monster SLayer fail, can be gleaned from this section of their story, also I believe taken from Reichard:
When The Twins visited Sun the second time, he said he was willing to help them, but this time he wanted them to return the favor: “I wish you to send your mother to the west that she may make a new home for me.” Whereupon Monster Slayer, believing himself equal to any task, replied, “I will do so.I will send her there.” Then Child-of-the-water reminded them both: “No, Changing Woman is subject to no one? we cannot make promises for her. She must speak for herself? she is her own mistress. But I shall tell her your wishes and plead for you.”
One commentator glibly suggests that the joint presentation of the hero as twins is “a clever reminder that progress depends upon cooperation between our mind and our heart” — but the psychologist Dr Howard Teich offers a far more depthful interpretation: that the two twins represent two forms of masculine heroism, one the familiar macho hero of war movies, and the other wiser and subtler, the possessor of traits commonly attributed to the feminine — and hugely undervalued — in our culture.
Dr Teich suggests we must (urgently) abandon the division of virtues into “male” and “female” types, reognize that these types are complementary rather than rivalrous, that both are necessary functions of both males’ and females’ psyches, and begin to integrate the wholeness that both strategies together represent, in our own approaches to our lives in general, to the natural world around us, and indeed to warfare — unsurprisingly, since we first encounter the twins in the ceremonial specifically devised by the Navajo to protect young warriors on their way to battle, and to reintegrate them in harmony and balance on their return.
As Teich puts it:
Monster Slayer and Child Born of Water, as these Twin Heroes are called, are the most sacred of all the legendary heroes in Navaho mythology. It is rare for the Navaho even to speak of the twins; their presence is to be felt rather than observed, and their lessons absorbed rather than applied. Although the lessons the twins hold may be countless, their particular manifestation of a deeper, more complex image of masculinity deserves the reader’s especial attention.
I’d like to suggest that in the same way that there are “zooming in” and “zooming out” styles in mathematics, and “monster-slayer” and “born of water” styles of heroism, there are in fact twin traditions of understanding the world which we might term scientific and poetic, or in Teich’s terms — and those of the alchemists — solar and lunar.
A unified or “solunary” vision will encompass the virtues of both.
Dr Teich’s review of the King > Oakes > Campbell book under the title A Dual Masculinity was irst piublished in The San Francisco Jung Institute Library Journal, Vol. 13, No. 4, 1995. He now has a book out treating these themes: Solar Light, Lunar Light.
Oh, and please don’t expect me to know anything more about Grothendieck’s mathematics than I read in Landsburg’s article.